Premium
Eigenvalue analysis of a block Red‐Black Gauss–Seidel preconditioner applied to the Hermite collocation discretization of Poisson's equation
Author(s) -
Brill Stephen H.,
Pinder George F.
Publication year - 2001
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.2
Subject(s) - mathematics , preconditioner , collocation (remote sensing) , discretization , orthogonal collocation , dirichlet boundary condition , collocation method , poisson's equation , gauss–seidel method , mathematical analysis , boundary value problem , linear system , differential equation , iterative method , mathematical optimization , ordinary differential equation , computer science , machine learning
Abstract This article is concerned with the numerical solution of Poisson's equation with Dirichlet boundary conditions, defined on the unit square, discretized by Hermite collocation with uniform mesh. In [1], it was demonstrated that the Bi‐CGSTAB method of van der Vorst [2] with block Red‐Black Gauss–Seidel (RBGS) preconditioner is an efficient method to solve this problem. In this article, we derive analytic formulae for the eigenvalues that control the rate at which the Bi‐CGSTAB/RBGS method converges. These formulae, which depend upon the location of the collocation points, can be utilized to determine where the collocation points should be placed in order to make the Bi‐CGSTAB/RBGS method converge as quickly as possible. Furthermore, using the optimal location of the collocation points can result in significant time savings for fixed accuracy and fixed problem size. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 204–228, 2001